Question

The image shows the first few steps of the polynomial division (3x4+9x2−13)÷(x2−5x)
. Complete the division process and find the remainder of this polynomial division.

(1 point)
Responses

The remainder is 2,087.
The remainder is 2,087.

The remainder is 407.
The remainder is 407.

The remainder is 420x−13
.
The remainder is 420 x minus 13 .

The remainder is 407x
.
The remainder is 407 x .

Answer 1

To complete the polynomial division \( (3x^4 + 9x^2 - 13) \div (x^2 - 5x) \), we will perform long division step by step.

1. **Divide the leading term**:
\( \frac{3x^4}{x^2} = 3x^2 \)

2. **Multiply** \( 3x^2 \) by \( (x^2 - 5x) \):
\[
3x^2 \cdot (x^2 - 5x) = 3x^4 - 15x^3
\]

3. **Subtract**:
\[
(3x^4 + 9x^2 - 13) - (3x^4 - 15x^3) = 15x^3 + 9x^2 - 13
\]

4. **Repeat the process**:
Divide the leading term:
\( \frac{15x^3}{x^2} = 15x \)

5. **Multiply**:
\[
15x \cdot (x^2 - 5x) = 15x^3 - 75x^2
\]

6. **Subtract**:
\[
(15x^3 + 9x^2 - 13) - (15x^3 - 75x^2) = 84x^2 - 13
\]

7. **Repeat the process**:
Divide the leading term:
\( \frac{84x^2}{x^2} = 84 \)

8. **Multiply**:
\[
84 \cdot (x^2 - 5x) = 84x^2 - 420x
\]

9. **Subtract**:
\[
(84x^2 - 13) - (84x^2 - 420x) = 420x - 13
\]

At this point, the degree of the remainder \( (420x - 13) \) is less than the degree of the divisor \( (x^2 - 5x) \), which means we cannot divide anymore.

Therefore, the final result of the division is:
\[
\text{Quotient} = 3x^2 + 15x + 84
\]
\[
\text{Remainder} = 420x - 13
\]

Given the options provided, the correct response is:
**The remainder is \( 420x - 13 \)**.